Double Degeneracy in Multiphase Modulation and the Emergence of the Boussinesq Equation

2018

Stud Appl Math

Self Cite

This paper illustrates how the singularity of the wave action flux causes the Kadomtsev‐Petviashvili (KP) equation to arise naturally from the modulation of a two‐phased wavetrain, causing the dispersion to emerge from the classical Whitham modulation theory. Interestingly, the coefficients of the resulting KP are shown to be related to the associated conservation of wave action for the original wavetrain, and therefore may be obtained prior to the modulation. This provides a universal form for the KP as a dispersive reduction from any Lagrangian with the appropriate wave action flux singularity. The theory is applied to the full water wave problem with two layers of stratification, illustrating how the KP equation arises from the modulation of a uniform flow state and how its coefficients may be extracted from the system.

“…The particular case this paper concerns itself with is when the wave action flux B is no longer injective in k . This means that there exists some curve

k = {boldk}_{0} false(boldm, ω false)

along which one has

\det [{normalD}_{boldk} B] = 0 .

Section: Abstract Setupmentioning

confidence: 97%

Section: Conservation Of Wave Actionmentioning

confidence: 98%

\begin{matrix} true \\ right Z false(x, y, t false) & = & left \overset{Z}{true} false(θ + ε φ (X, Y, T; ε) + ε^{2} ψ (X, Y, T; ε); \\ left boldk + ε^{2} boldq (X, Y, T; ε) + ε^{3} boldp (X, Y, T; ε), \\ left boldm + ε^{3} boldr (X, Y, T; ε) + ε^{4} bolds (X, Y, T; ε), \\ left ω + ε^{4} Ω (X, Y, T; ε) + ε^{5} Γ (X, Y, T; ε) false) \\ left + ε^{3} W (θ, X, Y, T; ε), \end{matrix}

for

X = ε x, Y = ε^{2} y, T = ε^{3} t

with the following phase relations:

\begin{matrix} true \\ right φ_{X} \end{matrix}

…”

Section: Modulation Leading To the Kp Equationmentioning

confidence: 99%

Φ

boldΦ = bold-italicφ (X, Y, T; ε) + ε bold-italicψ (X, Y, T; ε),

inspired by the latter multiphase works.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Flux singularities in multiphase wavetrains and the Kadomtsev‐Petviashvili equation with applications to stratified hydrodynamics

2018

Stud Appl Math

Self Cite

Nonlinear Theory for Coalescing Characteristics in Multiphase Whitham Modulation Theory

Bridges

2020

J Nonlinear Sci

Self Cite

The multiphase Whitham modulation equations with N phases have 2N characteristics which may be of hyperbolic or elliptic type. In this paper, a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly, a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation, that is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling wave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.

Phase Dynamics of the Dysthe Equation and the Bifurcation of Plane Waves

2019

Water Waves

The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such occurrences may be described using a Korteweg -de Vries (KdV) equation. The solitary wave solutions of this system form a qualitative prototype for the bifurcating dynamics, and the role of mean flow and steepening is then made clear through how they enter the amplitude and width of these solitary waves. Additionally, higher order phase dynamics are investigated, leading to increased nonlinear regimes which in turn have a more profound impact on how the plane waves transform under defects in the phase.